The dissipative ratio always increases as far asIn order to have stable background evolution one requires \(\Upsilon \propto T^\kappa \)  with \(\kappa \le 4\) , \(f_T \le 5\) , i.e., we may transfer energy to the thermal bath at a rate slower than the redshifting of standard radiation [1]}. Once \(\kappa > 4\)  radiation will become dominant and inflation ends. \(f_T < 5\)  , and so does \(T/H\) , but \(\epsilon \)  decreases when \(Q > (5-f_T)/2(1- f_T)\) , and again increases when \(1 < f_T \le 5\)  for whatever value of \(Q\) . That means that to end inflation in the strong dissipative regime, \(Q \gg 1\) , we must be already in the low \(T\)  regime with \(f_T > 1\)  and \(\Upsilon \propto T^\kappa \) , \(\kappa =f_T-1 >0\) . An example of this behaviour is given in Fig. (REF ), where we have plotted the evolution of the dissipative ratio \(Q\) , \(T/H\) , \(\epsilon _H\simeq \epsilon \) , \(gM/T\)  and \(d \ln \Upsilon / d \ln T = f_T -1\) , with respect to the no. of efolds left to the end of inflation, taken as \(\epsilon _H=1\) . As an example, we have numerically integrated the background equations for the parameter values \(M= 10^{-4} m_P\) , \(g=1\) , \(h=2.5\) , \(h_S=4\) , \(\lambda =10^{-14}\) , and \(g_*=12\) . At 50-60 e-folds before the end, we are in the high \(T\)  but weak dissipative regime, and \(\epsilon _H \simeq \epsilon _\phi /(1 +Q)\)  increases. When \(Q \gtrsim 5/2\)  the slow-parameter starts decreasing, but soon after \(gM/T>1\) , we move into the low \(T\)  regime, \(\epsilon _H\)  increases again and inflation ends.
<FIGURE>